Assessing the sustainability performance of manufacturing systems has mostly focused on the Triple Bottom Line dimensions. However, little emphasis has been placed on assessing circularity together with sustainability and in highlighting the tradeoffs that may exist between different assessment dimensions. This study presents a generic method aimed at facilitating integrated assessment of circularity, environmental and social sustainability specifically focused on manufacturing systems. The method consists of a list of indicators defined for each of the three assessment dimensions, classified in key categories. The categories’ importance is obtained through a survey with experts from the manufacturing sector. Subsequently, the survey outcomes are processed using the Best-Worst Method, a multi-criteria decision-making method, to obtain the weights for each indicator category. A case study application, serving as a worked example of the method, is presented to facilitate understanding of its operation and utility. The method can support effective assessment of manufacturing systems’ sustainability performance, while at the same time considering circularity performance and highlighting the potential tradeoffs.